= Quadratic drag on a sphere with constant radial growth
{title2=$v(t)\sim\sqrt{8\rho g\mu t/(3C_D\rho_a)}$}
For <surface-fed growth of an accreting sphere> with $\mu=\dot r>0$, stationary incoming material, and <quadratic drag> $F_D=C_D\rho_a\pi r^2v^2/2$, the <variable-mass system> obeys $\dot v+3\mu v/r=g-3C_D\rho_av^2/(8\rho r)$. For constant positive parameters, $r\sim\mu t$ and the late balance gives the displayed <speed>. The omitted acceleration and accretion terms are both $O(t^{-1/2})$, consistent with the balance. Thus this growing sphere has unbounded <speed> with vanishing <acceleration>, rather than a finite <terminal velocity>. This model has a different growth law from the existing <accreting raindrop with quadratic drag>, whose growth rate is proportional to speed.
Back to article page
